The homology of moduli spaces of 4-manifolds may be infinitely generated
For a simply-connected closed manifold X of $\dim X \neq 4$ , the mapping class group $\pi _0(\mathrm {Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifold...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050862400026X/type/journal_article |
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| Summary: | For a simply-connected closed manifold X of
$\dim X \neq 4$
, the mapping class group
$\pi _0(\mathrm {Diff}(X))$
is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each
$k>0$
, we prove that there are simply-connected closed smooth 4-manifolds X for which
$H_k(B\mathrm {Diff}(X);\mathbb {Z})$
are not finitely generated. The infinitely generated subgroup of
$H_k(B\mathrm {Diff}(X);\mathbb {Z})$
which we detect are topologically trivial, and unstable under the connected sum of
$S^2 \times S^2$
. These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory. |
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| ISSN: | 2050-5086 |